Hopf algebras arising from partial (co)actions

In this paper, extending the idea presented by Takeuchi in [M. Takeuchi, Matched pairs of groups and bismash products of Hopf algebras, Comm. Algebra9 (1981) 841-882.] and more generally by Majid in [S. Majid, Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction, J. Algebra130(1) (1990) 17-64.], we introduce the notion of partial matched pair (H,L) involving the concepts of partial action and partial coaction between two bialgebras H and L. Furthermore, we present sufficient conditions for the corresponding bismash product L#H to generate a new Hopf algebra and, as illustration, a family of examples is provided. Moreover, under some hypotheses such sufficient conditions are also necessary conditions.